INCREASING LIMIT OF REPRODUCING KERNELS

We discuss the limit of increasing reproducing kernels and construct the corresponding reproducing kernel Hilbert space. Given an increasing sequence of reproducing kernel Hilbert spaces with their norms decreasing, we can find a limit of the sequence of those reproducing kernels. Thereby we can construct a reproducing kernel Hilbert space on a subset of common underlying sets. The proof corrects an error in that of Aronszajn. We discuss the error by giving an example.